Some results on the commutativity of prime near-rings using generalized derivations

Abstract

Given a right near ring N, an additive mapping D: N →N is said to be a derivation on N, if D(xy) = D(x)y + xD(y) for all x, y ∈ N and an additive mapping F : N → N satisfying F(xy) = F(x)y + xD(y) for all x, y ∈ N, is called generalized derivation on N associated with the derivation D. The aim of this paper is to study the commutativity of a near-ring using some properties of generalized derivations on the given near-ring and we proved the following results. (a) The commutativity of 3-torsion free prime near-ring N with a generalized derivation F associated with non-zero idempotent derivation D on N satisfying the conditions F²[x, y] - [x, y] = 0 for all x, y ∈ N and F²(xoy) - (xoy) = 0 for all x, y ∈ N and (b) commutativity of 5-torsion free prime near ring N with a generalized derivation F associated with non-zero idempotent derivation D on N satisfying the conditions F²[x, y] + [x, y] = 0 for all x, y ∈ N and F²(xoy) + (xoy) = 0 for all x, y ∈ N are proved in this article. These results may help us to study more about the commutativity of general near-rings.